Solve for $x$ : $ 2|x + 3| + 2 = 1|x + 3| + 6 $
Subtract $ {1|x + 3|} $ from both sides: $ \begin{eqnarray} 2|x + 3| + 2 &=& 1|x + 3| + 6 \\ \\ { - 1|x + 3|} && { - 1|x + 3|} \\ \\ 1|x + 3| + 2 &=& 6 \end{eqnarray} $ Subtract ${2}$ from both sides: $ \begin{eqnarray} 1|x + 3| + 2 &=& 6 \\ \\ { - 2} &=& { - 2} \\ \\ 1|x + 3| &=& 4 \end{eqnarray} $ Simplify: $ |x + 3| = 4$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 3 = -4 $ or $ x + 3 = 4 $ Solve for the solution where $x + 3$ is negative: $ x + 3 = -4 $ Subtract ${3}$ from both sides: $ \begin{eqnarray} x + 3 &=& -4 \\ \\ {- 3} && {- 3} \\ \\ x &=& -4 - 3 \end{eqnarray} $ $ x = -7 $ Then calculate the solution where $x + 3$ is positive: $ x + 3 = 4 $ Subtract ${3}$ from both sides: $ \begin{eqnarray} x + 3 &=& 4 \\ \\ {- 3} && {- 3} \\ \\ x &=& 4 - 3 \end{eqnarray} $ $ x = 1 $ Thus, the correct answer is $x = -7 $ or $x = 1 $.